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Singular Structures on Liquid Rims Hans C March 2014 Volume 26 Number 3 Physics of Fluids pof.aip.org PHYSICS OF FLUIDS 26, 032109 (2014) Singular structures on liquid rims Hans C. Mayer and Rouslan Krechetnikov Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA (Received 5 November 2013; accepted 27 February 2014; published online 28 March 2014) This experimental note is concerned with a new effect we discovered in the course of studying water hammering phenomena. Namely, the ejecta originating from the solid plate impact on a water surface brings about a liquid rim at its edge with the fluid flowing towards the rim center and forming a singular structure resembling a “pancake.” Here, we present the experimental observations and a qualitative physical explanation for the effect, which proves to be fundamental to the situation when the size and speed of the impacting body are such that the capillary effects become important. C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4868730] The interplay of the continuous and the discrete is intrinsic in the description of the world – the emergence of discrete structures from continuous data has always been fascinating to scientists not only for its visual appeal but also for its fundamental importance.1 In the present work, we will discuss a particular exhibition of such phenomena due to the competition between inertia and surface tension effects. Namely, in the course of water hammering experiments, cf. Figure 1, we observed a phenomenon which resembles a “pancake” formed on liquid rims, a well-developed case of which is illustrated in Figure 2. Its formation in time is shown in Figure 3, where one can observe how the ejecta grows, a liquid rim forms at its end due to retraction, and finally a pancake appears. There are a few key questions originating from these observations. First, (I) what makes the fluid flow towards the center of the rim thus leading to the formation of the pancake structure? Second, we observed that the pancake preferrably forms along the shorter side of the impacting plate, which brings the related questions as to (II) why does ejecta develop higher along the long side of the impactor compared to that along the shorter side of the impactor as one can infer from Figure 3(a)? Also, (III) why does the ejecta develop higher along the sides of the impactor compared to the ejecta originating from under the corner as can be seen from both Figures 2 and 3? All these questions stimulated a more systematic study of this interesting phenomenon to be discussed below. For the measurements, we used rectangular impactor plates with the same long side of 2 l = 63.5 mm, but the short side was varied in the range w = 15.9 − 28.6 mm; the impact depths explored were in the range h = 0.5 − 2 mm and the impact velocities V0 = 380 − 700 mm/s. In order to find the transitions between the “no pancake” and “pancake” regimes, we fixed the impactor width and impact depth and then spun the impact velocity. While visually it is obvious what one may call a “pancake,” cf. Figure 2, for a systematic analysis of the data we called the structure in the middle of the rim a “pancake” if its diameter is at least 1.5 that of the rim and if by the time of its formation at least half of it is above the undisturbed water level. The results of our findings are summarized in Figure 4, which shows the transition measurements in the space defined by the impact velocity V0 and impactor width w. Two types of transitions – “no pancake to pancake” (np-p) and “pancake to corrugations” (p-c) – were identified as illustrated in Figure 4(a) for impact depth 1 mm. Figure 4(b) reflects the effect of the impact depth on the transition from “no pancake to pancake.” There are a few key trends to point out. Generally, in the order of increasing impact speed V0, one goes through the regimes of “no pancake,” “pancake,” and “corrugations,” cf. Figure 4(a); the increase in the depth of impact h shifts the transition from “no pancake to pancake” to lower values of V0, cf. Figure 4(b). As one can see in Figure 4(a),the 1070-6631/2014/26(3)/032109/9/$30.00 26, 032109-1 C 2014 AIP Publishing LLC This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.89.186.101 On: Fri, 28 Mar 2014 15:26:41 032109-2 H. C. Mayer and R. Krechetnikov Phys. Fluids 26, 032109 (2014) FIG. 1. Experimental apparatus used to produce “pancake” structures on liquid rims. The impacting plate of width w and length 2 l >wis connected to the plunger of a solenoid. When the solenoid is actuated by a laboratory power supply, the plate is set into motion and impacts the liquid surface with a speed V0 (adjusted through the power supply voltage) creating the splash (shown as dashed lines). The depth of impact h can be varied via the support structure. Impact events are filmed with a high-speed digital camera (Phantom v5.2) and side lighting from a LED source (IDT). pancake formation takes place over a narrow range of velocities, in between the transition from “no pancake to pancake” and “pancake to corrugations,” which explains the rarity of the phenomenon. Next, the difference between the liquid rim radius at the apex of the ejecta evolution versus the maximum pancake diameter over the entire course of the ejecta evolution as functions of the impact velocity V0 is shown in Figure 5. Figure 5(a) suggests that the rim diameter at the apex of ejecta elevation increases with the impact depth, and Figure 5(b) indicates that for a fixed depth h the rim diameter is not a strong function of the impact speed V0 and plate width w, but for deeper impacts, e.g., depth of 2 mm, the wider plates lead to larger pancakes, cf. Figure 5(a). With all these data reported, the idea now is to develop basic physical intuition about the observed trends summarized above as well as to answer the key questions (I)–(III). First, note that based on the incompressible pressure-impulse theory,2 the gauge pressure-impulse distribution + √ 0 = ρ 2 − 2 = across the plate is given by −0 p dt V0 l x , where x 0 corresponds to the plate center O and x = l to the plate edge S, cf. Figure 6. Thus, the pressure gradient driving fluid from under FIG. 2. Sequence of events leading to the pancake (46.7 ms after the impact) formed along the shorter side of the impactor of size 25.4 × 63.5 mm (impact depth 2 mm and velocity V0 = 528 mm/s). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.89.186.101 On: Fri, 28 Mar 2014 15:26:41 032109-3 H. C. Mayer and R. Krechetnikov Phys. Fluids 26, 032109 (2014) (a) (b) FIG. 3. Time sequence of side (a) and front (b) views of the “pancake” structure evolution (the impact conditions are given in the caption of Figure 2). the center O of the impactor to the edge is higher for the fluid flowing towards the long side of the impactor L compared to the shorter side S, which entails differences in the ejecta velocity and thus answers question (II). Similarly (question (III)), in part the ejecta raises more along the impactor sides because the pressure gradient driving fluid (wall jet) from under the center of the impactor is This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.89.186.101 On: Fri, 28 Mar 2014 15:26:41 032109-4 H. C. Mayer and R. Krechetnikov Phys. Fluids 26, 032109 (2014) np (a) 700 np-p p p-c s] / [mm 600 0 V 500 impact velocity, 400 15 20 25 30 plate width, w [mm] (b) 550 s] / [mm 500 0 V 450 h =0.5mm impact velocity, h =1.0mm 400 h =1.5mm h =2.0mm 15 20 25 30 plate width, w [ mm ] FIG. 4. Measurements of the transition (np-p) from “no pancake to pancake” in the space defined by the impact velocity V0 versus impactor width w: (a) two types of transitions – “no pancake to pancake” (np-p) and “pancake to corrugations” (p-c) for impact depth h = 1.0 mm; (b) effect of the impact depth h on the transition from “no pancake to pancake” (np-p) – deeper impacts shift the transition to lower velocities V0. The dashed and dotted lines are provided only to guide the eye. higher for the fluid flowing towards the edges S, L of the impactor compared to that towards the corner C. Also, one may think in terms of the high curvature κ of the corner and its analogy to the impact of a small radius κ−1 disk, which would not produce as much ejecta due to surface tension σ ρ 2/ κσ domination over inertia, V0 ( ) 1. Finally, concerning question (I), since the velocity at the corner Vc is lower compared to that at the center Vs , the pressure at the corner is higher, pc > ps, and thus there must be fluid flow towards the center of the edge S and hence up along the rim, cf.
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